3.12.44 \(\int \frac {2+3 x^5}{(-1+x^2+x^5) \sqrt [3]{-x+x^6}} \, dx\)

Optimal. Leaf size=85 \[ -\log \left (\sqrt [3]{x^6-x}+x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^6-x} x+\left (x^6-x\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 1.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 3*x^5)/((-1 + x^2 + x^5)*(-x + x^6)^(1/3)),x]

[Out]

(9*x*(1 - x^5)^(1/3)*Hypergeometric2F1[2/15, 1/3, 17/15, x^5])/(2*(-x + x^6)^(1/3)) + (15*x^(1/3)*(-1 + x^5)^(
1/3)*Defer[Subst][Defer[Int][x/((-1 + x^15)^(1/3)*(-1 + x^6 + x^15)), x], x, x^(1/3)])/(-x + x^6)^(1/3) - (9*x
^(1/3)*(-1 + x^5)^(1/3)*Defer[Subst][Defer[Int][x^7/((-1 + x^15)^(1/3)*(-1 + x^6 + x^15)), x], x, x^(1/3)])/(-
x + x^6)^(1/3)

Rubi steps

\begin {align*} \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \int \frac {2+3 x^5}{\sqrt [3]{x} \sqrt [3]{-1+x^5} \left (-1+x^2+x^5\right )} \, dx}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x \left (2+3 x^{15}\right )}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {3 x}{\sqrt [3]{-1+x^{15}}}+\frac {x \left (5-3 x^6\right )}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x \left (5-3 x^6\right )}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}+\frac {\left (9 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (9 \sqrt [3]{x} \sqrt [3]{1-x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {5 x}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )}-\frac {3 x^7}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {9 x \sqrt [3]{1-x^5} \, _2F_1\left (\frac {2}{15},\frac {1}{3};\frac {17}{15};x^5\right )}{2 \sqrt [3]{-x+x^6}}-\frac {\left (9 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}+\frac {\left (15 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(2 + 3*x^5)/((-1 + x^2 + x^5)*(-x + x^6)^(1/3)),x]

[Out]

Integrate[(2 + 3*x^5)/((-1 + x^2 + x^5)*(-x + x^6)^(1/3)), x]

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IntegrateAlgebraic [A]  time = 2.68, size = 85, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x+x^6}}\right )-\log \left (x+\sqrt [3]{-x+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x+x^6}+\left (-x+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + 3*x^5)/((-1 + x^2 + x^5)*(-x + x^6)^(1/3)),x]

[Out]

-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-x + x^6)^(1/3))]) - Log[x + (-x + x^6)^(1/3)] + Log[x^2 - x*(-x + x^6)^
(1/3) + (-x + x^6)^(2/3)]/2

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fricas [A]  time = 3.21, size = 104, normalized size = 1.22 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{6} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} - 1\right )} + 2 \, \sqrt {3} {\left (x^{6} - x\right )}^{\frac {2}{3}}}{x^{5} - 8 \, x^{2} - 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} + x^{2} + 3 \, {\left (x^{6} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{6} - x\right )}^{\frac {2}{3}} - 1}{x^{5} + x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+2)/(x^5+x^2-1)/(x^6-x)^(1/3),x, algorithm="fricas")

[Out]

-sqrt(3)*arctan((4*sqrt(3)*(x^6 - x)^(1/3)*x + sqrt(3)*(x^5 - 1) + 2*sqrt(3)*(x^6 - x)^(2/3))/(x^5 - 8*x^2 - 1
)) - 1/2*log((x^5 + x^2 + 3*(x^6 - x)^(1/3)*x + 3*(x^6 - x)^(2/3) - 1)/(x^5 + x^2 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + 2}{{\left (x^{6} - x\right )}^{\frac {1}{3}} {\left (x^{5} + x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+2)/(x^5+x^2-1)/(x^6-x)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x^5 + 2)/((x^6 - x)^(1/3)*(x^5 + x^2 - 1)), x)

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maple [C]  time = 7.68, size = 539, normalized size = 6.34

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {62685010802979296884 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-494135918029415995819 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}+64534903374029948502 x^{5}-485808833723089550851 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+433300799797487350553 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {2}{3}}-433300799797487350553 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {1}{3}} x +487658726294140202469 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+427751122084335395699 \left (x^{6}-x \right )^{\frac {2}{3}}-427751122084335395699 x \left (x^{6}-x \right )^{\frac {1}{3}}-62685010802979296884 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-56207819067703503534 x^{2}+494135918029415995819 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-64534903374029948502}{x^{5}+x^{2}-1}\right )-\ln \left (\frac {-11150604486509769179735404 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-167804753076351270129207425 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-624632666348769715120455774 x^{5}+86417184770450711142949381 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+971392776987982024558606028 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {2}{3}}-971392776987982024558606028 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {1}{3}} x +706020234654670274106713818 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+434526704299398906631777541 \left (x^{6}-x \right )^{\frac {2}{3}}-434526704299398906631777541 x \left (x^{6}-x \right )^{\frac {1}{3}}+11150604486509769179735404 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-92538172792410328165993448 x^{2}+167804753076351270129207425 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+624632666348769715120455774}{x^{5}+x^{2}-1}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {-11150604486509769179735404 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-167804753076351270129207425 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-624632666348769715120455774 x^{5}+86417184770450711142949381 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+971392776987982024558606028 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {2}{3}}-971392776987982024558606028 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {1}{3}} x +706020234654670274106713818 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+434526704299398906631777541 \left (x^{6}-x \right )^{\frac {2}{3}}-434526704299398906631777541 x \left (x^{6}-x \right )^{\frac {1}{3}}+11150604486509769179735404 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-92538172792410328165993448 x^{2}+167804753076351270129207425 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+624632666348769715120455774}{x^{5}+x^{2}-1}\right )\)	\(539\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^5+2)/(x^5+x^2-1)/(x^6-x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^2-_Z+1)*ln((62685010802979296884*RootOf(_Z^2-_Z+1)^2*x^5-494135918029415995819*RootOf(_Z^2-_Z+1)*x^5
+64534903374029948502*x^5-485808833723089550851*RootOf(_Z^2-_Z+1)^2*x^2+433300799797487350553*RootOf(_Z^2-_Z+1
)*(x^6-x)^(2/3)-433300799797487350553*RootOf(_Z^2-_Z+1)*(x^6-x)^(1/3)*x+487658726294140202469*RootOf(_Z^2-_Z+1
)*x^2+427751122084335395699*(x^6-x)^(2/3)-427751122084335395699*x*(x^6-x)^(1/3)-62685010802979296884*RootOf(_Z
^2-_Z+1)^2-56207819067703503534*x^2+494135918029415995819*RootOf(_Z^2-_Z+1)-64534903374029948502)/(x^5+x^2-1))
-ln((-11150604486509769179735404*RootOf(_Z^2-_Z+1)^2*x^5-167804753076351270129207425*RootOf(_Z^2-_Z+1)*x^5-624
632666348769715120455774*x^5+86417184770450711142949381*RootOf(_Z^2-_Z+1)^2*x^2+971392776987982024558606028*Ro
otOf(_Z^2-_Z+1)*(x^6-x)^(2/3)-971392776987982024558606028*RootOf(_Z^2-_Z+1)*(x^6-x)^(1/3)*x+706020234654670274
106713818*RootOf(_Z^2-_Z+1)*x^2+434526704299398906631777541*(x^6-x)^(2/3)-434526704299398906631777541*x*(x^6-x
)^(1/3)+11150604486509769179735404*RootOf(_Z^2-_Z+1)^2-92538172792410328165993448*x^2+167804753076351270129207
425*RootOf(_Z^2-_Z+1)+624632666348769715120455774)/(x^5+x^2-1))*RootOf(_Z^2-_Z+1)+ln((-11150604486509769179735
404*RootOf(_Z^2-_Z+1)^2*x^5-167804753076351270129207425*RootOf(_Z^2-_Z+1)*x^5-624632666348769715120455774*x^5+
86417184770450711142949381*RootOf(_Z^2-_Z+1)^2*x^2+971392776987982024558606028*RootOf(_Z^2-_Z+1)*(x^6-x)^(2/3)
-971392776987982024558606028*RootOf(_Z^2-_Z+1)*(x^6-x)^(1/3)*x+706020234654670274106713818*RootOf(_Z^2-_Z+1)*x
^2+434526704299398906631777541*(x^6-x)^(2/3)-434526704299398906631777541*x*(x^6-x)^(1/3)+111506044865097691797
35404*RootOf(_Z^2-_Z+1)^2-92538172792410328165993448*x^2+167804753076351270129207425*RootOf(_Z^2-_Z+1)+6246326
66348769715120455774)/(x^5+x^2-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + 2}{{\left (x^{6} - x\right )}^{\frac {1}{3}} {\left (x^{5} + x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+2)/(x^5+x^2-1)/(x^6-x)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x^5 + 2)/((x^6 - x)^(1/3)*(x^5 + x^2 - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x^5+2}{{\left (x^6-x\right )}^{1/3}\,\left (x^5+x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^5 + 2)/((x^6 - x)^(1/3)*(x^2 + x^5 - 1)),x)

[Out]

int((3*x^5 + 2)/((x^6 - x)^(1/3)*(x^2 + x^5 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{5} + 2}{\sqrt [3]{x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (x^{5} + x^{2} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**5+2)/(x**5+x**2-1)/(x**6-x)**(1/3),x)

[Out]

Integral((3*x**5 + 2)/((x*(x - 1)*(x**4 + x**3 + x**2 + x + 1))**(1/3)*(x**5 + x**2 - 1)), x)

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